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Applying compressive sensing (CS) allows for sub-Nyquist sampling in severalapplication areas in 5G and beyond. This reduces the associated training,feedback, and computation overheads in many applications. However, theapplicability of CS relies on the validity of a signal sparsity assumption andknowing the exact sparsity level. It is customary to assume a foreknownsparsity level. Still, this assumption is not valid in practice, especiallywhen applying learned dictionaries as sparsifying transforms. The problem ismore strongly pronounced with multidimensional sparsity. In this paper, wepropose an algorithm for estimating the composite sparsity lying in multipledomains defined by learned dictionaries. The proposed algorithm estimates thesparsity level over a dictionary by inferring it from its counterpart withrespect to a compact discrete Fourier basis. This inference is achieved by amachine learning prediction. This setting learns the intrinsic relationshipbetween the columns of a dictionary and those of such a fixed basis. Theproposed algorithm is applied to estimating sparsity levels in wirelesschannels, and in cognitive radio spectra. Extensive simulations validate a highquality of sparsity estimation leading to performances very close to theimpractical case of assuming known sparsity.

Estimating Sparsity Level for Enabling Compressive Sensing of Wireless Channels and Spectra in 5G and Beyond